Diagonally drift-implicit Runge-Kutta methods of weak order one and two for Itô SDEs and stability analysis. (English) Zbl 1166.65304
Summary: The class of stochastic Runge-Kutta methods for stochastic differential equations due to A. Rößler [BIT 46, No. 1, 97–110 (2006; Zbl 1091.65004); SIAM J. Numer. Anal. 47, No. 3, 1713–1738 (2009)] is considered. Coefficient families of diagonally drift-implicit stochastic Runge-Kutta (DDISRK) methods of weak order one and two are calculated. Their asymptotic stability as well as mean-square stability (MS-stability) properties are studied for a linear stochastic test equation with multiplicative noise. The stability functions for the DDISRK methods are determined and their domains of stability are compared to the corresponding domain of stability of the considered test equation. Stability regions are presented for various coefficients of the families of DDISRK methods in order to determine step size restrictions such that the numerical approximation reproduces the characteristics of the solution process.
MSC:
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 65C20 | Probabilistic models, generic numerical methods in probability and statistics |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
Keywords:
asymptotic stability; mean-square stability; stochastic Runge-Kutta method; implicit method; stochastic differential equation; weak approximationCitations:
Zbl 1091.65004References:
| [1] | Burrage, K.; Burrage, P. M.; Mitsui, T., Numerical solutions of stochastic differential equations – implementation and stability issues, J. Comput. Appl. Math., 125, 171-182 (2000) · Zbl 0971.65003 |
| [2] | Burrage, K.; Tian, T., Implicit stochastic Runge-Kutta methods for stochastic differential equations, BIT, 44, 21-39 (2004) · Zbl 1048.65005 |
| [3] | Debrabant, K.; Kværnø, A., B-Series analysis of stochastic Runge-Kutta methods that use an iterative scheme to compute their internal stage values, SIAM J. Numer. Anal., 47, 1, 181-203 (2008) · Zbl 1188.65006 |
| [4] | Debrabant, K.; Rößler, A., Families of efficient second order Runge-Kutta methods for the weak approximation of Itô stochastic differential equations, Appl. Numer. Math., 59, 3-4, 582-594 (2009) · Zbl 1166.65303 |
| [5] | Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II (1996), Springer-Verlag: Springer-Verlag Berlin · Zbl 0859.65067 |
| [6] | Has’minskii, R. Z., Stochastic Stability of Differential Equations (1980), Sijthoff & Noordhoff: Sijthoff & Noordhoff Alphen aan den Rijn, Germantown · Zbl 0276.60059 |
| [7] | Hernandez, D. B.; Spigler, R., \(A\)-stability of Runge-Kutta methods for systems with additive noise, BIT, 32, 4, 620-633 (1992) · Zbl 0755.65084 |
| [8] | Hernandez, D. B.; Spigler, R., Convergence and stability of implicit Runge-Kutta methods for systems with multiplicative noise, BIT, 33, 4, 654-669 (1993) · Zbl 0789.65101 |
| [9] | Higham, D. J., Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal., 38, 3, 753-769 (2000) · Zbl 0982.60051 |
| [10] | Hofmann, N.; Platen, E., Stability of weak numerical schemes for stochastic differential equations, Comput. Math. Appl., 28, 10-12, 45-57 (1994) · Zbl 0810.65147 |
| [11] | Kloeden, P. E.; Platen, E., Numerical Solution of Stochastic Differential Equations, Applications of Mathematics, vol. 23 (1999), Springer-Verlag: Springer-Verlag Berlin · Zbl 0701.60054 |
| [12] | Y. Komori, Weak first- or second-order implicit Runge-Kutta methods for stochastic differential equations with a scalar Wiener process, J. Comput. Appl. Math. (2007), doi:10.1016/j.cam.2007.06.024; Y. Komori, Weak first- or second-order implicit Runge-Kutta methods for stochastic differential equations with a scalar Wiener process, J. Comput. Appl. Math. (2007), doi:10.1016/j.cam.2007.06.024 · Zbl 1147.65004 |
| [13] | Komori, Y.; Mitsui, T., Stable ROW-type weak scheme for stochastic differential equations, Monte Carlo Methods Appl., 1, 4, 279-300 (1995) · Zbl 0841.65146 |
| [14] | Milstein, G. N.; Platen, E.; Schurz, H., Balanced implicit methods for stiff stochastic systems, SIAM J. Numer. Anal., 35, 3, 1010-1019 (1998) · Zbl 0914.65143 |
| [15] | Milstein, G. N.; Tretyakov, M. V., Stochastic Numerics for Mathematical Physics, Scientific Computation (2004), Springer-Verlag: Springer-Verlag Berlin · Zbl 1059.65007 |
| [16] | Peterson, W. P., A general implicit splitting for stabilizing numerical simulations of Itô stochastic differential equations, SIAM J. Numer. Anal., 35, 1439-1451 (1998) · Zbl 0914.65144 |
| [17] | Rößler, A., Rooted tree analysis for order conditions of stochastic Runge-Kutta methods for the weak approximation of stochastic differential equations, Stochastic Anal. Appl., 24, 1, 97-134 (2006) · Zbl 1094.65008 |
| [18] | Rößler, A., Runge-Kutta methods for Itô stochastic differential equations with scalar noise, BIT, 46, 1, 97-110 (2006) · Zbl 1091.65004 |
| [19] | Rößler, A., Runge-Kutta methods for Stratonovich stochastic differential equation systems with commutative noise, J. Comput. Appl. Math., 164-165, 613-627 (2004) · Zbl 1038.65003 |
| [20] | A. Rößler, Second order Runge-Kutta methods for Itô stochastic differential equations, Preprint No. 2479, Technische Universität Darmstadt, 2006; A. Rößler, Second order Runge-Kutta methods for Itô stochastic differential equations, Preprint No. 2479, Technische Universität Darmstadt, 2006 |
| [21] | Rößler, A., Second order Runge-Kutta methods for Stratonovich stochastic differential equations, BIT, 47, 3, 657-680 (2007) · Zbl 1127.65007 |
| [22] | Saito, Y.; Mitsui, T., T-stability of numerical scheme for stochastic differential equations, World Sci. Ser. Appl. Anal., 2, 333-344 (1993) · Zbl 0834.65146 |
| [23] | Saito, Y.; Mitsui, T., Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33, 6, 2254-2267 (1996) · Zbl 0869.60052 |
| [24] | Schurz, H., Asymptotical mean square stability of an equilibrium point of some linear numerical solutions with multiplicative noise, Stochastic Anal. Appl., 14, 313-354 (1996) · Zbl 0860.60040 |
| [25] | Schurz, H., The invariance of asymptotic laws of linear stochastic systems under discretization, Z. Angew. Math. Mech., 79, 6, 375-382 (1999) · Zbl 0938.34072 |
| [26] | Tocino, A., Mean-square stability of second-order Runge-Kutta methods for stochastic differential equations, J. Comput. Appl. Math., 175, 355-367 (2005) · Zbl 1063.65009 |
| [27] | Tocino, A.; Vigo-Aguiar, J., Weak second order conditions for stochastic Runge-Kutta methods, SIAM J. Sci. Comput., 24, 2, 507-523 (2002) · Zbl 1025.65011 |
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